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Three-step design optimization for multi-channel fibre Bragg gratings

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Abstract

Methods to produce optimal designs for multi-channel fiber Bragg gratings (FBGs) with identical or close to identical channel-to-channel spectral characteristics are discussed. The proposed approach consists of three distinct steps. The first two steps (preliminary semi-analytic minimization and subsequent fine-tuning) do not depend on the grating design details, but on the number of channels only and can be readily applied to similar problems in other fields, e.g., in radio-physics and coding theory. The third step (spectral characteristic quality improvement) is FBG field specific. A comparison with other known optimization methods shows that the proposed approach yields generally superior results for small to moderate number of channels (N<60).

©2003 Optical Society of America

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Figures (3)

Fig. 1.
Fig. 1. Normalized peak index change as a result of the first two steps in the three-step optimization process. Solid curve shows the analytic estimate (Eq. (7)).
Fig. 2.
Fig. 2. An illustration of three-stage optimization of a 9-channel dispersion compensator design. Non-trivially modulated phase profile and group delay characteristics are not shown. (a), (b) the amplitude profile and the transmission spectrum obtained after the first step of optimization; (c), (d) the same as (a), (b) but after the second step; (e), (f) result of the third step. The final result is presented in more detail in Fig. 3.
Fig. 3.
Fig. 3. Details of the 9-channel dispersion compensator design shown in Figs. 2(e,f). (a) amplitude and phase profiles; (b) enlarged (a); (c) central part of the reflection spectrum; (d) group delay. We note that, all fast oscillations of κ(z) profile in the vicinity of the main peak (z≈2.6) were completely eliminated after 30 iterations, though some unimportant weak modulation of the profile in the region z≈3.8 still presents.

Equations (15)

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E b z + i δ E b q ( z ) E f = 0 ,
E f z i δ E f q * ( z ) E b = 0 ,
[ E b ( 0 ) E f ( 0 ) ] = [ 1 t * r 1 t r 2 t 1 t ] [ E b ( L ) E f ( L ) ] ,
q ( z ) = l = 1 N κ ( z ) e i θ ( z ) e i [ ( 2 l N 1 ) Δ k z 2 + ϕ l ] = κ ( z ) e i θ ( z ) S ( z ) ,
arg { S ( z ) } = arctan [ l = 1 N sin ( [ 2 l N 1 ] Δ k z 2 + ϕ l ) l = 1 N cos ( [ 2 l N 1 ] Δ k z 2 + ϕ l ) ] .
S ( z ) = N ( 1 + 2 N Re p = 1 N 1 C p e i p Δ k z ) 1 2 ,
C P = l = 1 N p m l + p m l * , p = 1 , 2 , , N 1 ,
Δ n N = max z S ( z ) Δ n 1 = N + 2 Δ n 1 ,
Δ ( ϕ ) = [ s ( z ) s ( z ) ] 2 = 1 s ( z ) 2 ,
s ( z ) = 1 1 4 N 2 p = 1 N 1 C p 2 + O ( x 3 ) .
Δ n env ( av ) = N ( 1 1 4 N 2 ) Δ n 1 .
F = 1 2 N + O ( 1 N 2 ) .
Δ ( ϕ ) = 1 2 N p = 1 N 1 C p 2 + O ( x 3 x 2 ) ,
Δ ( ϕ ) ϕ l = Im { p = 1 l 1 C p m l p m l * + p = 1 N l C p * m l + p m l * } 2 N 2 Δ ( ϕ ) + O ( x 3 x 2 ) ,
1 2 q ( z 2 ) = + r ( δ ) exp ( i δ z ) d δ .
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